Quantum numbers are fundamental, discrete properties that characterize particles and arise from the symmetries of quantum systems. They include the **Principal Quantum Number** \( n \), which specifies an electron's energy level or shell in an atom, and the **Orbital Angular Momentum Quantum Number** \( l \), which describes the shape of the orbital within that shell. Additionally, the **Magnetic Quantum Number** \( m_l \) defines the orientation of the orbital in space, while the **Spin Quantum Number** \( s \) represents the intrinsic spin of a particle. Related to spin, the **Magnetic Spin Quantum Number** \( m_s \) describes the specific orientation of this intrinsic spin in a magnetic field. Moving beyond electron structure, the **Isospin Quantum Number** \( I \) and **Isospin Projection** \( I_3 \) are used to describe nucleons in the context of the strong nuclear force, providing a symmetry between protons and neutrons. Other quantum numbers reflect the presence of specific quark flavors within particles, such as **Strangeness** \( S \), **Charm** \( C \), **Bottomness** \( B' \), and **Topness** \( T \). Particles are also assigned a **Baryon Number** \( B \) if they are baryons (e.g., protons or neutrons) and a **Lepton Number** \( L \) if they are leptons (e.g., electrons or neutrinos).
Further quantum numbers include **Parity** \( P \), which describes how a particle behaves under spatial inversion, and **Charge Conjugation** \( C \), which indicates its transformation properties under particle-antiparticle exchange. **G-Parity** combines charge conjugation with isospin rotation for mesons, while **Hypercharge** \( Y \), defined as the sum of baryon number and strangeness, plays a role in strong interactions. Finally, **Total Angular Momentum** \( J \) combines orbital and spin angular momentum, and **Weak Isospin** \( T \) or \( T_3 \) relates to weak force interactions. These quantum numbers are crucial for identifying and classifying particles as well as determining the possible interactions they can undergo, as they are often conserved in processes.
Beyond these quantum numbers, particles also exhibit other properties that are not quantized or conserved in the same way but are still essential for understanding quantum behavior. The **wavefunction** \(\psi\) encapsulates information about the probability amplitude of a particle’s state, while **phase** represents relative differences that affect interference patterns in quantum systems. **Coherence** is another critical property, indicating the correlation between quantum states' phases and enabling phenomena like superposition and entanglement. **Entanglement** itself describes a deep, non-local connection between particles, where the state of one particle depends on another, yet this relationship is not a quantum number. Particles with spin also have a **spin orientation** in space that affects interactions but is not captured by the spin quantum number alone.
Certain properties, like **parity violation** in weak interactions, describe how symmetries might be broken in processes without implying a specific conserved value. **Chirality** and **helicity** further describe "handedness" and the alignment of spin with momentum, respectively, particularly relevant for massless particles, though they are not quantum numbers. Other important particle attributes include **mass** and **energy**, which are fundamental but continuous variables. **Decay modes and lifetimes** characterize particle stability and transformation but do not represent discrete quantum values. Properties like **cross-sections** and **scattering amplitudes** measure interaction probabilities, while **gauge symmetries** and **coupling constants** (defining interaction strengths) set up interaction frameworks rather than particle-specific quantum states. Collectively, these properties play crucial roles in quantum behavior and interaction dynamics but remain distinct from quantum numbers in their non-discrete or environment-dependent nature.